How Many Microhenrys in That Coil?
December 1931/January 1932 Short Wave Craft

Dec. 1931 / Jan. 1932 Short Wave Craft [Table of Contents] Wax nostalgic about and learn from the history of early electronics. See articles from Short Wave Craft, published 1930 - 1936. All copyrights hereby acknowledged.

This is a nice short article covering the calculation of inductances for coils wound on cores and wire sizes. It appeared in a 1932 issue of Short Wave Craft magazine, but of course inductance has not changed since then so it is still relevant. The author recognized that standard formulas, although concise and accurate, are sometimes difficult to work with when calculations for a large number of values is needed for a particular circuit design. To address the situation, he presents a handy nomograph, chart, and a table of typical values. He also introduces a rarely seen term "Nagaoka's correction factor*" for skin effect. A smartphone app, a spreadsheet, or a desktop computer program would be used today to calculate inductance values, number of turns, winding spacing, etc., but back when mechanical slide rules were the order of the day, these visual methods were a real benefit.

* "The Inductance Coefficients of Solenoids," Hantaro Nagaoka, 1909

How Many Microhenrys in That Coil?

By James K. Clapp*

Every radio student should know how to calculate the inductance of a coil of given or known size. Here's a simplified method worked out by a leading engineer.

While much material has been published on the calculation of the inductance of coils†. the formulae given are in general not convenient for engineering use. Two difficulties are encountered in applying the results in engineering practice, one being the involved computations and the other the fact that differences in form and wire sizes and errors in the measurement of these factors introduce errors in the calculations which largely vitiate the utility of precise formulae.

For single-layer coils at radio frequencies (and, with slight modification, for bank-wound coils), Nagaoka's formula probably is the best for general engineering use. While neglecting the shape and size of the cross-section of the wire, the self-capacity of the winding and the variation of inductance due to skin-effect, it may be shown that the formula gives about as good results for high-frequency inductance as can be obtained.

Tables of the values of Nagaoka's correction factor have been prepared, but require considerable time to use due to the necessity for interpolations. The table values may be plotted in the form of a curve, but a more convenient interpolation is made possible by plotting these values on logarithmic scales, as has been done in Figure 3. Where much work of this type is done, the scales may be transferred to a slide-rule so that no reference to printed material is required.

The formulae given here, when carefully applied, give values of inductance to within about two per cent for single-layer coils and to within about five per cent for four-layer bank-wound coils for frequencies where the coils would serve as normal tuned-circuit elements.

The general formula is

where a is radius of a mean turn in inches, n is the number of turns, b is the length of the winding in inches, and K is Nagaoka's correction factor which is a function of  or the ratio of diameter to length of the winding.

If n₀ is the number of turns per inch, the inductance and ratio of diameter to length are more conveniently given by:

L = 0.1003 * a² * n * n₀ * K, microhenrys (2)

or L = 0.0251 * d² * n * n₀ * K, microhenrys (3)

where   numeric (4)

and d is the diameter of the mean turn in inches.

Given the size of wire and its insulation and the diameter of the coil form, n₀ as wound, is found from Table I and is readily computed for any desired number of turns. Read the corresponding value of K from the scales at the left. The inductance is then easily computed by means of the slide-rule.

For banked windings of not too great depth as compared with the diameter, a close approximation for the inductance is obtained by using Nn₀ for the turns per inch (where N is the number of banks) and for the ratio of diameter to length.

Then = numeric (5) and L = 0.0251 * d² * N * n * n₀ * K, microhenries        (6)

The number of turns required for a desired value of inductance cannot be directly calculated since K varies as n is varied. With given types of windings experience will indicate an approximate value for the number of turns. If the computations are carried out and the inductance obtained is near the desired value, the correct number of turns to give the desired value may be obtained by readjustment, since K does not vary rapidly with n. Where many values are required it is simpler to calculate a sufficient number of values for a curve. The required values may then be read off directly. (See Figures 1 and 2, for example.)

Examples of Calculations

Given: Form diameter = 2.75 inches (General Radio Company Type 577 Form). Wire size = No. 20 double-silk-covered. Find: The inductance for coil of 35 turns.

Procedure: In Table 1 find n₀ = 25

From scales, opposite 1.99 for , read

K= 0.526

L = 0.0251 * (2.79)² * 35 * 25 * 0.526 = 90.0 microhenries.

For a rough estimate, the diameter of the form may often be taken as the diameter of a turn. In the above example this procedure gives = 1.965, K = 0.530 and L = 88 microhenries, which differs from the previous value by about 2.5 per cent.

For bank-wound coils an example is as follows:

Given: d = 2.75, n₀ =25, N = 4, and n = 200

Then = 1.455.

From Figure 3, K = 0.604

Then 4 x 25 x 200 x 0.604 = 2570 microhenries.

Many experimenters and many engineers "design" inductors by guessing at the number of turns, then peeling off wire until the correct value of inductance is obtained rather than go to the trouble of using the usual tables and formulas. Our experience with the method described here proves conclusively that much time and effort are saved by calculating the desired value of inductance before the coil is wound. - Courtesy "General Radio Experimenter."

*Engineer, General Radio Company

†See in particular the publications of the U. S. Bureau of Standards and the Proceedings of the Institute of Radio Engineers.

Nomographs / Nomograms Available on RF Cafe:

- Symmetrical Reactance Chart

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- Symmetrical T and H Attenuator Nomograph

- Amplifier Gain Nomograph

- Decibel Nomograph

- Voltage and Power Level Nomograph

- Nomograph Construction

- Nomogram Construction for Charts with Complicating Factors or Constants

- Link Coupling Nomogram

- Multi-Layer Coil Nomograph

- Delay Line Nomogram

- Voltage, Current, Resistance, and Power Nomograph

- Resistor Selection Nomogram

- Resistance and Capacitance Nomograph

- Capacitance Nomograph

- Earth Curvature Nomograph

- Coil Winding Nomogram

- RC Time-Constant Nomogram

- Coil Design Nomograph

- Voltage, Power, and Decibel Nomograph

- Coil Inductance Nomograph

- Antenna Gain Nomograph

- Resistance and Reactance Nomograph

- Frequency / Reactance Nomograph